Problem: Factor the following expression: $-9$ $x^2+$ $34$ $x$ $-21$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-9)}{(-21)} &=& 189 \\ {a} + {b} &=& & & {34} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $189$ and add them together. The factors that add up to ${34}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${27}$ $ \begin{eqnarray} {ab} &=& ({7})({27}) &=& 189 \\ {a} + {b} &=& {7} + {27} &=& 34 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-9}x^2 +{7}x +{27}x {-21} $ Group the terms so that there is a common factor in each group: $ ({-9}x^2 +{7}x) + ({27}x {-21}) $ Factor out the common factors: $ x(-9x + 7) - 3(-9x + 7) $ Notice how $(-9x + 7)$ has become a common factor. Factor this out to find the answer. $(-9x + 7)(x - 3)$